Question: Factor the following expression: $-2$ $x^2+$ $7$ $x+$ $4$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-2)}{(4)} &=& -8 \\ {a} + {b} &=& & & {7} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-8$ and add them together. Remember, since $-8$ is negative, one of the factors must be negative. The factors that add up to ${7}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${8}$ $ \begin{eqnarray} {ab} &=& ({-1})({8}) &=& -8 \\ {a} + {b} &=& {-1} + {8} &=& 7 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-2}x^2 {-1}x +{8}x +{4} $ Group the terms so that there is a common factor in each group: $ ({-2}x^2 {-1}x) + ({8}x +{4}) $ Factor out the common factors: $ x(-2x - 1) - 4(-2x - 1) $ Notice how $(-2x - 1)$ has become a common factor. Factor this out to find the answer. $(-2x - 1)(x - 4)$